Fracture mechanics, Damage and Fatigue Linear Elastic ... · PDF fileUniversity of...

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University of Liège Aerospace & Mechanical Engineering Fracture mechanics, Damage and Fatigue Linear Elastic Fracture Mechanics Crack Growth Fracture Mechanics LEFM Crack Growth Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 http://www.ltas-cm3.ulg.ac.be/ Chemin des Chevreuils 1, B4000 Liège [email protected]

Transcript of Fracture mechanics, Damage and Fatigue Linear Elastic ... · PDF fileUniversity of...

Page 1: Fracture mechanics, Damage and Fatigue Linear Elastic ... · PDF fileUniversity of Liège Aerospace & Mechanical Engineering Fracture mechanics, Damage and Fatigue Linear Elastic Fracture

University of Liège

Aerospace & Mechanical Engineering

Fracture mechanics, Damage and Fatigue

Linear Elastic Fracture Mechanics – Crack Growth

Fracture Mechanics – LEFM – Crack Growth

Ludovic Noels

Computational & Multiscale Mechanics of Materials – CM3

http://www.ltas-cm3.ulg.ac.be/

Chemin des Chevreuils 1, B4000 Liège

[email protected]

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Elastic fracture

• Definition of elastic fracture

– Strictly speaking:

• During elastic fracture, the only changes to the material are atomic separations

– As it never happens, the pragmatic definition is

• The process zone, which is the region where the inelastic deformations

– Plastic flow,

– Micro-fractures,

– Void growth, …

happen, is a small region compared to the specimen size, and is at the crack tip

• Therefore

– Linear elastic stress analysis describes the fracture process with accuracy

Mode I Mode II Mode III

(opening) (sliding) (shearing)

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• SIFs (KI, KII, KIII) define the asymptotic solution in linear elasticity

• Crack closure integral

– Energy required to close the crack by an infinitesimal da

– If an internal potential exists

with

• AND if linear elasticity

– AND if straight ahead growth

• J integral – Energy that flows toward crack tip

– If an internal potential exists

• Is path independent if the contour G embeds a straight crack tip

• BUT no assumption on subsequent growth direction

• If crack grows straight ahead G=J

• If linear elasticity:

• Can be extended to plasticity if no unloading (see later)

Summary

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Elastic fracture in mode I

• Crack growth criterion in mode I

– For a crack to grow

• The energy released by the system has to be larger than the fracture energy Gc

• Criterion:

• The energy release rate G depends on

– The geometry

– The loading

– The crack size

• What does the fracture energy Gc depend on ?

– For perfectly brittle materials: this is a material dependant constant

– Practically, it has to account for the inelastic deformations happening in the

process zone

– For (an)isotropic materials, Gc is (not) the same in all the directions

» Example, in wood cracks grow along grains

– We consider isotropic linear elastic materials

– For an initial straight crack under mode I: • The crack will grow straight ahead (see next slides)

• So toughness and fracture energy are related

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Elastic fracture in mode I

• Crack growth criterion in mode I (2)

– Toughness is defined for plane e state

• It is conservative as

• Near the free surface

– Deformations along thickness

release the stress state

– The SIF is lower and

– Process zone is not negligible

(see lecture on NLFM)

Gc is larger near free surfaces

• Plane strain & elastic fracture can be assumed

if the working zone is small compared

to the specimen and the crack length

– Thick specimen

(see lecture on NLFM)

– Crack large enough

(see lecture on NLFM)

t

Kc

K rupture

Plane s

Plane e

Brittle fracture

t1

Ductile

fracture t2<<t1

Plane e

t

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Elastic fracture in mode I

• Crack growth criterion in mode I (3)

– Toughness & fracture energy of brittle materials

Material KC [MPa · m½] GC [J · m-2]

Borosilicate Glass 0.8 9.

Alumina 99% polycrystalline 4. 39.

Zirconia-Toughened Alumina 6. 90.

Yttria Partially Stabilized Zirconia 13. 730.

Aluminum 7075-T6 25. 7800.

AlSiC Metal Matrix Composite 10. 400.

Epoxy 0.4 200.

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Elastic fracture in mode I

• Crack growth criterion in mode I (4) – Behavior depends on environmental conditions

• Example: DBTT – High strength steel alloys – Toughness divided by 3 – Fracture energy divided by 9

brittle

ductile

• We know if the crack will grow

• But – How fast ?

– How far ?

– In which direction ?

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Crack grow stability

• Considering a cracked body with G ≥ GC: the crack will grow

– If GC is constant: two possible behaviors as the crack grows

• G decreases the crack stops growing (unless the loading increases)

• G increases instability and the specimen fractures

– If GC is not constant: the question becomes

• Which one of G or GC will grow at the highest rate

– The stability of a crack growth depends on

• The geometry

• The loading

• The material behavior

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Crack grow stability

• Example: Delamination of composite (DCB specimen)

– Compliance (Linear elasticity)

– Prescribed loading

• As the crack is growing: G increases

• For a perfectly brittle material GC is constant

unstable

– Prescribed displacement

• As the crack is growing: G decreases

• For a perfectly brittle material GC is constant

stable

– Is it a general rule? What happens for a general loading?

a

h

h

Q

Thickness t

a

h

h

u

Thickness t

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Crack grow stability

• General loading conditions

– Practically, when a crack propagates

• Part of the structure becomes more compliant

• Part of the load is transferred to other parts

neither fixed load nor fixed displacement

– This corresponds to a compliant machine loading

• Spring of compliance CM

• Generalized loading Q(a) (spring and cracked body)

• Generalized displacement at the cracked body u(a)

• Prescribed displacement uT at one extremity of the spring Q(a)

u(a)

C(a)

uT

Q(a)

CM

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Crack grow stability

• General loading conditions (2)

– Variation of the energy release rate

• Dead load

– Spring of infinite compliance (CM → ∞)

– As first term is <0, the value CM → ∞

is the one for which ∂AG is maximum

• As the spring stiffens (toward a fixed grip)

– CM decreases

– ∂AG also decreases, which stabilizes the crack growth

– If ∂AG becomes <0, for a perfectly brittle material

(GC=constant) the crack becomes stable

– A fixed grip is always more stable than a dead load

• Fixed grip

• Dead load

Q(a)

u(a)

C(a)

uT

Q(a)

CM

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Crack grow stability

• Non-perfectly brittle material: Resistance curve

– A crack can be stable even if ∂AG > 0

– For non-perfectly brittle materials GC depends on the crack surface

• Therefore GC will be renamed the resistance Rc (A)

• In 2D: Rc (DA)= Rc (t Da)

• Elastoplastic behavior

– In front of the crack tip

there is an active plastic

zone

– Resistance increases when

the crack propagates as

the required plastic work

increases

– This effect is more

important for thin

specimens as the

elasto-plastic behavior

is more pronounced in plane s

– A steady state can be reached (active zone in a plastic wake) or not

– Composites: • As the crack propagates fibers in the wake tend to close crack tip

more and more energy is required for the crack to grow

Plastic

zone

Blunting

Initial crack

growth

Steady state

DA=tDa

Rc

Rc fragile

Rc ducile t1

0

Rc ducile t2<t1

Rc ducile t3<< (Plane s)

Active

plastic zone Plastic wake

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Crack grow stability

• Example: Delamination of composite with initial crack a0

Dead load vs Fixed grip

– Dead load Q1

• Perfectly brittle materials: unstable

• Ductile materials: stable, but if a is larger than a** it turns unstable

– Dead load Q2 > Q1

• This is the limit of stability for ductile materials (always unstable for perfectly brittle material)

– For Q3: unstable

a

Rc, G

Rc ducile

u increasing

a0

GC brittle

u1

u2

u3

a

Rc, G

Rc ducile

Q increasing

a0 a* a**

GC brittle

Q1

Q2 Q3

– Fixed grip u1 ,u2 or u3

• The crack is stable for any material

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Crack grow stability

• Crack growth stability criterion

– Crack growth criterion is G ≥ GC

– Stability of the crack is reformulated (in 2D)

• Stable crack growth if

• Unstable crack growth if

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Mixed mode fracture

• In which direction will the crack grow?

– For anisotropic & isotropic material

– Under mixed mode loading

– In composite

• Cracks follow the path of least resistance

– Important to predict this path during the design

– “Fail safe design”

• In the following we assume homogeneous & isotropic materials under

mixed mode loading

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Mixed mode fracture

• Combination of mode I & mode II loadings

– The crack will propagate with a kink angle q

– q=0 only if it corresponds to a weak plane of the material (bond line in

composites e.g.)

– Loadings in mode I and mode II

• Rotation

• Mode I loading (infinite plate):

• Mode II loading (infinite plate):

– How can we determine the kink angle?

2a

x

y

s∞

s∞

b q

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Mixed mode fracture

• Method of the maximum circumferential stress

– Erdogan & Sih, 1963

– Assumptions:

• The crack will grow in the direction where

the SIF related to mode I in the new frame

is maximal

• It will grow only if this SIF is larger than

the toughness of pure mode I loading

– So the new criterion is

with &

2a

x

y

s∞

s∞

b q* err eqq

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Mixed mode fracture

• Method of the maximum circumferential stress (2)

– Mode I

– Mode II

– Mixed mode I & mode II loadings

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Mixed mode fracture

• Method of the maximum circumferential stress (3)

– Mixed mode I & mode II loadings in new frame

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Mixed mode fracture

• Method of the maximum circumferential stress (4)

– Maximum hoop stress

• Direction of loading defined by

• In case of the infinite plate b* corresponds to b

• The maximum hoop stress becomes

2a

x

y

s∞

s∞

b*

q* err eqq

– For b*=90°:q*=0, which

corresponds to a straight

ahead growth under pure

mode I

– For b*>0 the maximum hoop

stress is reached for q*<0

-100 0 100-4

-2

0

2

4

q [°]

(2

r)1

/2 s

qq /

KI

b*=15°

b*=30°

b*=45°

b*=60°

b*=90°

q*

Maximum is for q <0

q [deg.]

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Mixed mode fracture

• Method of the maximum circumferential stress (5)

– Kink angle obtained when hoop stress is maximum

• So the kink angle q* is reached for

• This prediction is in good agreement

with experimental results obtained

by Erdogan and Sih (1963)

0 20 40 60 800

20

40

60

80

b* [°]

- q*

AnalyticExperimental

b* [deg.]

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Mixed mode fracture

• Method of the maximum circumferential stress (6)

– Failure envelope

• Criterion

becomes

with

• Resolution of this linear system of

2 equations with 2 unknowns

leads to the failure envelope

• Toughness tests for different

loading directions gives

comparisons points

• Maximum circumferential stress

theory is conservative 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

KI / K

C

KII

/ K

C

AnalyticExperimental

KI / KC

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Mixed mode fracture

• Method of the maximum circumferential stress (7)

– Pure mode II theory (b * = 0, KI = 0)

• Kink angle q*|b*=0 ~ -70.53°

• Failure if KII = 0. 87KC

• N.B.: For pure mode II the case of a plate in tension is meaningless

– One should consider shearing

2a

x

y

s∞

s∞

b*

q* err eqq

2a

x

y

t∞

t∞

q* err eqq

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Mixed mode fracture

• Method of the maximum circumferential stress (8)

– Pure mode II: plate under shearing

• Circumferential stress:

– Maximum for q =-45°

• Theory predicts a kink angle q*|b*=0 ~ -70.53°

• Why is there a difference?

– Due to the stress field at crack tip

the maximum circumferential stress

is at -70.53° crack will grow

with this angle

– As it grows, the crack will not remain

under pure mode II loading

– Crack will turn until growing

with a -45-degree angle

– Second order theory

• For a plate in shearing it has been shown

that the kink |angle| will be reduced as the

crack propagates

• What is the general rule?

2a x

y t∞

t∞

2a x

y t∞

t∞

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Mixed mode fracture

• Method of the maximum circumferential stress (9)

– Second order theory (2)

• Order 0 has to be added to the asymptotic

solution (in 1/r1/2) when r ~ rc

• Asymptotic solution characterized by SIFs

• Order 0 characterized by T along Ox e.g.

• New kink angle q * (b*, T, rc, KI) depends on stress field at rc

– Useful for FE computation

– rc determined by experiment (Aluminum alloy: rc ~1.5 mm)

2a

x

y

T

T

b*

q* err eqq

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Mixed mode fracture

• Method of the maximum circumferential stress (10)

– Second order theory (3)

• Results obtained numerically can be summarized as: after initial kinking

– For compression (T < 0) or low traction: crack will tend to be aligned with T

– For traction of the order of KI /rc1/2 or larger: there is bifurcation

2a

2a 2a

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Mixed mode fracture

• Method of the maximum energy release rate

– In thermodynamics: equilibrium corresponds to the lowest potential energy

• So the crack will propagate to minimize the potential energy

• As the energy release rate is defined by

the crack will grow in the direction maximizing G

– Methodology

• ki: SIFs at the crack tip before kink propagation

• Ki: SIFs at extremity of a kink of infinitesimal length and angle a

• Ki(a) can be deduced from ki: (Cotterell & Rice, 1980)

– First order accurate in a

• As the kink grows straight ahead:

• Fracture criterion and growth direction are defined such that

, &

kI & kII a

KI & KII

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Mixed mode fracture

• Method of the maximum energy release rate (2)

– Loading direction defined by cot b* = kII/kI

– It gives results similar to the method of maximum hoop stress

2a

x

y

s∞

s∞

b

a* err eqq

-100 0 1000

5

10

15

20

25

a [°]

G( a

) /

G(0

)

b*=15°

b*=30°

b*=45°

b*=60°

b*=90°

a*

a [deg.]

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Mixed mode fracture

• Method of the maximum energy release rate (3)

– Comparison against the method of maximum hoop stress

• Recall we used a formula first order accurate in a

– < 5% error for a < 40°

– ~ 5% error on KI for a > 40° (Cotterell & Rice, 1980)

– For the method of maximum energy release rate:

• Before the crack propagates: kI & kII can be ≠ 0

• After kinking: KII = 0 the crack propagates under pure mode I

0 20 40 60 800

20

40

60

80

b* [°]

- q*

Max. hoopMax. G

b* [deg.]

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

KI / K

CK

II /

KC

Max. hoopK

i for Max. G

ki for Max. G

KI / KC

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Fatigue failure

• Under cyclic loadings a crack can

– Be initiated for loadings with s < sp0

– Propagate for Ki < KC

• Total life approaches

– Unable to account for inherent defects

• Example: design of the De Havilland Comet

using total life approach. Defects resulting

from punched riveting of square windows

caused failure of aircrafts

– Unable to predict crack propagation

P

P

a

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Fatigue failure

• Microscopic observations for cycling loading

– I. Crack initiated at stress concentrations (nucleation)

– II. Crack growth resulting into surface striations

– III. Failure of the structure when the crack reaches a critical size

– Example of a crank axis

• Development of damage tolerant design

– Assume cracks are present from the beginning of service

– Predict crack growth and end of life

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Fatigue failure

• I. Crack initiation

– Nucleation: cracks initiated for K << KC

• Surface: deformations result from

dislocations motion along slip planes

• Can also happen around a

bulk defect

P

P

a

N cycles

Pmin Pmax

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Fatigue failure

• II. Crack growth

– Stage I fatigue crack growth:

• Along a slip plane

– Stage II fatigue crack growth:

• Across several grains

– Along a slip plane in each grain

– Straight ahead macroscopically

• Striation of the failure surface:

corresponds to the cycles

• III. Structure failure – As the crack growth K tends toward KC

– For a critical size of the crack there is a static failure

Perfect crystal

Cra

ck g

row

th

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Fatigue failure

• Crack growth rate

– Tests: conditioning parameters

• DP &

• Pmin / Pmax

– SSY assumption

• As fatigue happens for

Ki < KC this assumption is

usually satisfied, at least

during crack growth stage

– Therefore, as SSY assumption

holds, fatigue failure can be

described uniquely by

• DK = Kmax- Kmin &

• R = Kmin/Kmax

– There is DKth such that if DK ~ DKth:

• The crack has a growth rate lower than one atomic spacing per cycle (statistical value)

• Dormant crack

Interval of striations

P

P

a

a (

cm)

Nf (105) 1 2 3 4 5 6 7

5

4

3

2

DP1

DP2 > DP1

Structure inspection

possible (for DP1) Rapid crack

growth

Rupture

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Fatigue failure

• Crack growth rate (2)

– Zone I

• Stage I fatigue crack growth

• DKth depends on R

– Zone II

• Stage II fatigue crack growth: striation

• 1963, Paris-Erdogan

– Depends on the geometry, the loading, the frequency

– Steel: DKth ~ 2-5 MPa m1/2 , C ~ 0.07-0.11 10-11 [m (MPa m1/2)-m], m ~ 4

– Steel in sea water: DKth ~ 1-1.5 MPa m1/2, C ~ 1.6 10-11 [idem], m ~ 3.3

• Be careful: K depends on a integration needed to get a(Nf)

– Infinite plate, mode I :

– Zone III

• Rapid crack growth until failure

• Static behavior (cleavage) due to the effect of Kmax(a)

• There is failure once af is reached, with af such that Kmax(af) = Kc

da

/ d

Nf

logDK DKth

Zone I Zone II Zone III

R

1

m

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Fatigue failure

• Experimental results

– Parameters in Paris law

Material DKth [MPa · m½] m [-] C [m (MPa . m1/2)-m]

Mild steel 3.2-6.6 3.3 0.24 . 10-11

Structural steel 2.0-5.0 3.85-4.2 0.07-0.11 . 10-11

Structural steel is sea water 1.0-1.5 3.3 1.6 . 10-11

Aluminum 1.0-2.0 2.9 4.56 . 10-11

Aluminum alloy 1.0-2.0 2.6-2.9 3-19 . 10-11

Copper 1.8-2.8 3.9 0.34 . 10-11

Titanium alloy (6Al-4V, R=0.1) 2.0-3.0 3.22 1 . 10-11

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Fatigue failure

• Effect of R=Kmin/Kmax: crack closure – During loading phases at maximum stress

• There is an active plastic zone (phase transformation can happen)

• The crack opening allows fluid or products to enter

– If R <0.7 or negative, at low stress crack lips can enter into contact due to

• Plasticity

– Residual plastic strain resulting

from plastic wake will close

the crack

• Roughness

– Non flat lips prevent sliding (mode II)

• Corrosion

– Corrosion products fill the opening

• Viscous fluid

– Lubricant fluids fill the opening

• Phase transformation

– Phase transformation at crack tip can put material

in compression

Plastic wake Active plastic zone

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Fatigue failure

• Effect of crack closure on fatigue

– When the loading decreases local compressive effects parts of the cracks are kept opened

– The stress intensity factor at the minimum of the cycle is more important than predicted the effective DK is actually reduced

– The crack closure effect is therefore beneficial to structure life

t

smax

smin

Ds = smax-smin

t

Kmax

Kmin

DK DKeff

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Fatigue failure

• Effect of R=Kmin/Kmax on threshold (Zone I)

– Due to Plasticity Induced Crack Closure (PICC)

– DKth decreases when R increases

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Fatigue failure

• Effect of R=Kmin/Kmax on crack growth rate (Zone II)

– Due to crack closure life of structure is improved for low R

• Example of 2024-T3 aluminum alloy*

– DKeff depends on many parameters (loading, environment, …)

• Example: model of Elber & Schijve for Al. 2024-T3

– DKeff = (0.55 + 0.33 R + 0.12 R2) DK for -1<R<0.54

• Models can be inaccurate in non-adequate circumstances

*J.C. Newman Jr, E.P. Phillips, M.H. Swain, Fatigue-life prediction methodology using small-crack theory, International Journal of Fatigue 21 (1999)

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Fatigue failure

• Overload effect

– What happens if there is a few (or a moderate) number of overloads ?

– Plastic wake is temporarily increased, until the active plastic zone at crack tip

passed the plastic zone created by the overload

– So which effect ?

Active plastic zone

t

Kmax

Kmin

DK

Plastic wake

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Fatigue failure

• Overload effect (2)

– As the plastic wake is temporarily increased, DKeff is reduced due to PICC

there is a retard effect in the crack propagation

• Infrequent overloads help

• Frequent overloads may help

• Too frequent overloads are

damaging, as it actually

corresponds to increasing Kmax

– There exist overload models

• Wheeler e.g.

– N.B. 1952, a fuselage of the Comet was tested against fatigue

• Static loading at 1.12 atm, followed by

• 10 000 cycles at 0.7 atm (> cabin pressurization at 0.58 atm)

• Production fuselage without the static loading failed after a few 1000 cycles

(pressurization at 0.58 atm)

a (

cm)

Nf (105) 1 2 3 4 5 6 7

5

4

3

2

Overloads

Retards

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Stress corrosion cracking

• A crack can grow due to the combination of stress and chemical attack

– This is not only fatigue but also for static stress with K<KC

– It happens in particular environments

• Salt water

• Hydrogen

• Chlorides

• …

– Mainly for metals

• Steel in salt water, chloride, hydrogen

• Aluminum alloys in salt water

Steel AISA

4335V

2013-2014 Fracture Mechanics – LEFM – Crack Growth 43

KI (MPa m1/2)

da/d

t (m

m/m

in)

10 100

10

1

0.1

0.01

H20

H2

Steel 4043

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• Double cantilever beam loaded with a compliant machine

– Assume a perfectly brittle material

– What is the effect of CM on

the crack growth stability?

Exercise 1

a

h

h

Thickness t

x

y

uT

CM

2013-2014 Fracture Mechanics – LEFM – Crack Growth 44

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• Edge notch specimen under cyclic loading – Assume titanium alloy 6%Al - 4%V

• See figures below

– Assume a remains << W and << h

– Initial crack size a = 1.5 cm

– Cyclic loading between • Minimum value: 8 MPA

• Maximum value: 80 MPa

– What is the life of the structure?

Exercise 2

y

x

s

s

W h

a

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• Improved materials to crack propagation – Periodical deflection of crack

• Depends on structure

– Using the model • Deflected crack with periodic tilts

• Crack opening/closing under cyclic

loading

– Establish

• SIF for growth along the kinks

• Effective SIF

• Apparent crack propagation rate

• Effect of surface mismatch

Exercise 3

Overaged (precipitation from heat treatment),

7475-T7351 aluminum alloy, R=0.1, vacuum

Underaged 7475 aluminum alloy, R=0.1, vacuum

S. Suresh, Metallurgica transactions A, 16A, 1985

S S

S

D D q

Dd

Dd*

uI uII

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• Improved materials to crack propagation (2) – Initial nanocrystalline material

• Subjected to cyclic mode I

• No crack deflection observed

• Crack growth data in table

– Grain size is increased to micro-order • Deflections observed

– Represent on a graph the crack growth data • For nanocrystalline materials

• For microcrystalline materials

Exercise 4

S. Suresh, Metallurgica transactions A, 16A, 1985

S S = 0.418 mm

S

D D = 0.255 mm q = /3

DK (MPa m1/2)

da/dN (mm/cycle)

2.573 2.32E-08

2.607 3.68E-08

2.6585 5.84E-08

2.7462 9.27E-08

2.8552 1.47E-07

3.0265 2.24E-07

3.2287 3.42E-07

3.4445 5.22E-07

3.7225 8.27E-07

3.9713 1.21E-06

4.2642 1.92E-06

4.6381 3.05E-06

4.98 4.48E-06

5.3474 7.10E-06

5.7415 1.04E-05

6.125 1.42E-05

6.4088 1.92E-05

6.9707 2.93E-05

7.4366 4.48E-05

2013-2014 Fracture Mechanics – LEFM – Crack Growth 47

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References

• Lecture notes

– Lecture Notes on Fracture Mechanics, Alan T. Zehnder, Cornell University,

Ithaca, http://hdl.handle.net/1813/3075

• Other references

– « on-line »

• Fracture Mechanics, Piet Schreurs, TUe, http://www.mate.tue.nl/~piet/edu/frm/sht/bmsht.html

– Book

• Fracture Mechanics: Fundamentals and applications, D. T. Anderson. CRC press, 1991.

• Fatigue of Materials, S. Suresh, Cambridge press, 1998.

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• Double cantilever beam loaded with a compliant machine

– Compliant machine

– DCB alone:

• Compliance

• Failure for a crack size a0 with

– Back to compliant machine

• Instability for a perfectly brittle material if if

or again if

• So for a perfectly brittle material it is unstable if CM > 2 C(a), but eventually as C

will grow with the crack, it will become larger than CM / 2 again

Exercise 1: Solution

a

h

h

Thickness t

x

y

uT

CM

2013-2014 Fracture Mechanics – LEFM – Crack Growth 49

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• Double cantilever beam loaded with a compliant machine

– Other method

– Normalizing with respect to the case CM=0

• Since it has been normalized

for critical uT of the case CM=0,

for a crack to grow we would

need a larger uT so G reaches GC

• If CM>2C(a0) there will be an

instability as for a/a0 =1 the slope

is positive

• Eventually, as the crack grows the slope

becomes negative again, stabilizing the crack

Exercise 1: Solution

2013-2014 Fracture Mechanics – LEFM – Crack Growth 50

0 1 2 30

0.05

0.1

0.15

0.2

0.25

a/a0

G/G

C

CM

/C=1/2

CM

/C=1

CM

/C=3/2

CM

/C=2

CM

/C=5/2

CM

/C=3

a /a0

G /

GC

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• Edge notch specimen under cyclic loading – Material properties at room temperature

• Yield: 830 MPa

• Toughness: 55 MPa · m½

– SIF if a remains < 2% of W

• KI = 1.122 s ( a)1/2

– Plane strain & elastic fracture?

• Yes if specimen thick enough

• Crack is large enough

• Moreover the applied stress << the yield stress

Exercise 2: Solution

y

x

s

s

W h

a

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• Edge notch specimen under cyclic loading (2)

– Is the crack critical?

• The crack will not lead to static failure

– Fatigue?

• As we are above the threshold there will be fatigue

• R = 0.1, so crack experiences closing effect

– What is the critical crack length leading to static failure?

Exercise 2: Solution

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• Edge notch specimen under cyclic loading (3)

– Parameters in Paris’ law

• There is a phase transition around

DK=17 MPa . m1/2 but we are above

• Paris’ coefficients:

– C=10-11 m (MPa . m1/2)-m

– m=3.22

Exercise 2: Solution

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• Edge notch specimen under cyclic loading (4)

– Paris’ law

• Can be integrated explicitly, and, for m≠2, it yields

• So the number of cycles in terms of the crack size is

– Critical size is reached after

1.74 105 cycles, but this value cannot

be used as

• Paris’ law is not valid is zone III

• After 1.5 105 cycles the crack is

growing too quickly for allowing

inspections

• 1.1 105 is a conservative time life

Exercise 2: Solution

2013-2014 Fracture Mechanics – LEFM – Crack Growth 54

0 0.5 1 1.5 2

x 105

0

0.05

0.1

Nf

a [

m]

N x 105

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• Edge notch specimen under cyclic loading (5)

– Some remarks

• If there is an error of 10% in the estimation

of the initial crack length size

– Life of the structure is

reduced by 15%

– Using a 1.5 105-cycle life prediction

would actually lead to a crack

size located in zone III

• We have assumed a < 0.02 W, which

corresponds to a six-meter wide

specimen. In practice

– Crack size cannot be considered

infinitely small

– SIF must then be evaluated using

» Either FEM simulations

» Or SIF handbooks

– Paris’ law

» Cannot be integrated in a closed form

» Integration has to be performed numerically

Exercise 2: Solution

2013-2014 Fracture Mechanics – LEFM – Crack Growth 55

0 0.5 1 1.5 2

x 105

0

0.05

0.1

Nf

a [

m]

a0=1.5 cm

a0=1.65 cm

a0=1.8 cm

N x 105

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• SIF for growth along the kinks

– Cortell & Rice

– For the problem under consideration

• Global mode I: kI = KIg & kII = 0

• Along kink: KI = KIk & KII = KII

k

Exercise 3: Solution

S. Suresh, Metallurgica transactions A, 16A, 1985

kI & kII a

KI & KII

S S

S

D D q

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• SIF for growth along the kinks (2)

– Global mode I: KIg

– Along kink: KIk & KII

k

Exercise 3: Solution

S. Suresh, Metallurgica transactions A, 16A, 1985

S S

S

D D q

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• Effective SIF

– Global mode corresponds to segment S: KIg

• As crack grows straight ahead on S :

– Along kink D: KIk & KII

k

• As crack grows straight ahead on D:

– Assumption: effective SIF is the weighted average of the SIFs

– For cyclic loading

Exercise 3: Solution

S. Suresh, Metallurgica transactions A, 16A, 1985

S S

S

D D q

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• Apparent crack propagation rate

– Effective SIF

– Crack growth

• Real: a = D+S

• Apparent: aapp = D cos q + S

– The 2 relations are the equations of LEFM for a periodically deflected crack

– But how to take into account contact during unloading?

Exercise 3: Solution

S. Suresh, Metallurgica transactions A, 16A, 1985

S S

S

D D q

aapp

Dd

Dd*

uI uII

Growth rate of a

straight crack

2013-2014 Fracture Mechanics – LEFM – Crack Growth 59

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• Effective SIF with surface mismatch

– Effective SIF without surface mismatch

– Surface mismatch under mode I opening

• During unloading: crack unloads in both mode I and II, leading to uII

– At first time of contact:

• Opening during peak loading is rewritten

• During peak loading, in LEFM:

• During closure:

Exercise 3: Solution

S. Suresh, Metallurgica transactions A, 16A, 1985

Dd

Dd*

uI uII

At peak load During unloading

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• Effective SIF with surface mismatch (2)

– Effective SIF without surface mismatch

– Due to surface mismatch under mode I opening

• As

• If

Exercise 3: Solution

S. Suresh, Metallurgica transactions A, 16A, 1985

Dd

Dd*

uI uII

At peak load During unloading

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• Microcrystalline materials

– Apparent crack growth rate

– Effective DK

Exercise 4: Solution

S S = 0.418 mm

S

D D = 0.255 mm q = /3

DK (MPa m1/2)

da/dN (mm/cycle)

2.573 2.32E-08

2.607 3.68E-08

2.6585 5.84E-08

2.7462 9.27E-08

2.8552 1.47E-07

3.0265 2.24E-07

3.2287 3.42E-07

3.4445 5.22E-07

3.7225 8.27E-07

3.9713 1.21E-06

4.2642 1.92E-06

4.6381 3.05E-06

4.98 4.48E-06

5.3474 7.10E-06

5.7415 1.04E-05

6.125 1.42E-05

6.4088 1.92E-05

6.9707 2.93E-05

7.4366 4.48E-05

S. Suresh, Metallurgica transactions A, 16A, 1985

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• Interpretation

– Data correspond to a straight crack:

• &

– What matters is the apparent propagation

• &

Exercise 4: Solution

DK (MPa m1/2)

da/dN (mm/cycle)

2.573 2.32E-08

2.607 3.68E-08

2.6585 5.84E-08

2.7462 9.27E-08

2.8552 1.47E-07

3.0265 2.24E-07

3.2287 3.42E-07

3.4445 5.22E-07

3.7225 8.27E-07

3.9713 1.21E-06

4.2642 1.92E-06

4.6381 3.05E-06

4.98 4.48E-06

5.3474 7.10E-06

5.7415 1.04E-05

6.125 1.42E-05

6.4088 1.92E-05

6.9707 2.93E-05

7.4366 4.48E-05

S. Suresh, Metallurgica transactions A, 16A, 1985

S S

S

D D q

aapp

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100

101

102

10-8

10-7

10-6

10-5

10-4

D K (MPa m1/2

)

da/

dN

(m

m/c

ycl

e)

nano

micro, =0.1

micro, =0.25

micro, =0.75

DK [Mpa m1/2]

• Graph

Exercise 4: Solution

DK (MPa m1/2)

da/dN (mm/cycle)

2.573 2.32E-08

2.607 3.68E-08

2.6585 5.84E-08

2.7462 9.27E-08

2.8552 1.47E-07

3.0265 2.24E-07

3.2287 3.42E-07

3.4445 5.22E-07

3.7225 8.27E-07

3.9713 1.21E-06

4.2642 1.92E-06

4.6381 3.05E-06

4.98 4.48E-06

5.3474 7.10E-06

5.7415 1.04E-05

6.125 1.42E-05

6.4088 1.92E-05

6.9707 2.93E-05

7.4366 4.48E-05

S. Suresh, Metallurgica transactions A, 16A, 1985

2013-2014 Fracture Mechanics – LEFM – Crack Growth 64